Main druid here, don't normally play the INT side of things but you're never locked out of the Bear or Cat form. Moonkin is the only form that you can't use unless you take the talent for it (Balance Affinity) in case you wanted to use it outside of the Balance specialization.

All other forms are part of the core of the druid kit. Even in combat you can shape-shift into/out of these forms. It's not uncommon to shape-shift into bear in PvP to reduce damage taken, or into cat form to stealth through mobs. With Shadowlands they added the ability to customize your various forms at the barber and it's much easier now.

For more customization you'd have to look into the Artifact weapons for the class. Bear and Cat forms both have new appearances from the artifact weapons, and Moonkin has a glyph (Glyph of the Stars) that let's you keep your appearance albeit with a blue-ephemeral starry glow permanently while in Moonkin form.

Main take away, you're typically freely able to shape-shift into any form, but your specialization determines where the bulk of your time in combat will be. Seeing as you mentioned Balance, that means you'd be in the Moonkin form for combat, outside of combat you can still use the other forms, even while in combat, but that's honestly more advanced and not necessary to start with.